The mean curvature flow is the gradient flow of the volume functional of submanifolds. The analytic nature is a parabolic system of nonlinear partial differential equations whose stationary phase corresponds to minimal submanifolds. Minimal hypersurfaces and mean curvature flows of hypersurfaces have been studied extensively for decades but many applications in mathematical physics and topology call for understanding of the higher codimensional case. This project proposes to embark on a systematic investigation of minimal submanifolds and mean curvature flows in higher codimensions. Three immediate goals are to understand the minimal surface system in higher codimension, the Lagrangian mean curvature flow in Calabi-Yau manifolds and the deformation of maps between Riemannian manifolds by the mean curvature flow.
Minimal surfaces are like soap films, they are surfaces of least area. The mean curvature flow is an evolution process which movesT a surface in space in such a way that its area is decreased most rapidly. This is a very natural yet highly nonlinear process and it models physics phenomena such as the motion of an interface in forming metallic alloys. Understanding the behavior of this evolution is important for simplifying and smoothing complicated surfaces in the most efficient way. This process, as a geometric evolution equation, tends to deform a geometric object to its optimal shape. A goal of this project is to apply this process to study certain special surfaces in Calabi-Yau manifolds, the spaces of string theory. It is believed that underlying structures of Calabi-Yau manifolds are encoded in these special surfaces, called special Lagrangians. The success of this proposal will have important impact on image processing, material science, mathematics physics and other nonlinear problems.
